Geodesic variation ($F(s,t),{\nabla }_t{\partial }_{t}F=0$)

A geodesic variation of a given geodesic is a smooth family $F(s,t)$ with $F(0,t)=\gamma (t)$, whose curves at fixed $s$ are affinely parametrized geodesics. Differentiating its geodesic equation and using the torsion-free Levi-Civita connection gives the Jacobi field equation $D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma }=0$ for $J={\partial }_{s}F{|}_{s=0}$.

Given a Jacobi field with initial values $J(0)=a$ and $D_{t}J(0)=b$, vary the initial point along a curve with tangent $a$ and the initial velocity with covariant derivative $b$. The resulting geodesic variation has the same Jacobi initial values and therefore the same field by uniqueness of the linear equation. Smooth dependence on initial conditions gives a common parameter interval around any fixed compact geodesic segment, even on an incomplete manifold.